Haiman's Conjecture and Springer's Representations

Minh-T\^am Quang Trinh

arxiv: 2605.20131 · v1 · pith:EFQWDUGVnew · submitted 2026-05-19 · 🧮 math.RT · math.AG· math.CO

Haiman's Conjecture and Springer's Representations

Minh-T\^am Quang Trinh This is my paper

Pith reviewed 2026-05-20 03:14 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.CO

keywords Haiman's conjectureSpringer representationsLusztig varietiesLLT polynomialsWeyl group characterspositivityunimodalityreductive groups

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The pith

The nonzero coefficients of α_ψ,G^z are positive and unimodal when ψ is inflated from type A.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the graded W-character of the intersection cohomology of closed Lusztig varieties for z over the regular semisimple locus of a reductive group G. It relates the resulting formula to unipotent Lusztig varieties, supplying a new geometric model for unicellular LLT polynomials. The authors then extract Laurent polynomials α_ψ,G^z that record how the character decomposes into ungraded irreducible characters coming from Springer theory. Low-rank calculations lead them to conjecture that these coefficients are positive and unimodal whenever ψ arises by inflating an irreducible character from a type-A Levi subgroup in the indicated manner. The same work proves that the matrix of all such α polynomials is partially triangular and that the positivity and unimodality properties are preserved under passage to Levi subgroups.

Core claim

Using work of Lusztig and Abreu-Nigro, we compute the graded W-character of the intersection cohomology of any closed Lusztig variety for z over the regular semisimple locus. Relating the formula to unipotent Lusztig varieties gives a new geometric model for unicellular LLT polynomials. We introduce Laurent polynomials α_ψ,G^z indexed by irreducible characters ψ that encode the decomposition into ungraded Springer characters. From evidence in low rank we conjecture that if ψ is inflated from type A in a particular way, then the nonzero coefficients of α_ψ,G^z are positive and unimodal. This offers an answer to a 1993 question of Haiman. We also prove that the matrix formed by the α_ψ,G^z is

What carries the argument

The Laurent polynomials α_ψ,G^z, which encode the decomposition of the graded W-character of the intersection cohomology into ungraded characters from Springer theory.

If this is right

The α_ψ,G^z supply a geometric model for unicellular LLT polynomials via unipotent Lusztig varieties.
The matrix whose entries are the α_ψ,G^z is partially triangular.
Positivity and unimodality of the nonzero coefficients are stable under inclusions of Levi subgroups.
The construction gives a concrete geometric approach to generalizing Haiman's conjecture beyond symmetric groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The geometric model may allow proofs of the positivity conjecture by exploiting properties of perverse sheaves or intersection cohomology.
The stability under Levi inclusions suggests that an inductive argument on rank could be used to establish the conjecture in all types.
Similar positivity statements might hold for other families of polynomials obtained from graded characters of Lusztig varieties.
Computational checks in rank four and five would give stronger evidence before attempting a general proof.

Load-bearing premise

The cited results of Lusztig and Abreu-Nigro apply directly to the intersection cohomology of closed Lusztig varieties over the regular semisimple locus in the present setting.

What would settle it

An explicit computation, for an irreducible character ψ inflated from type A in a group of rank three or higher, showing that at least one nonzero coefficient of the corresponding α_ψ,G^z is negative or that the sequence of coefficients is not unimodal.

read the original abstract

For any connected complex reductive group $G$ and element $z$ of its Weyl group $W$, we use work of Lusztig and Abreu-Nigro to compute the graded $W$-character of the intersection cohomology of any closed Lusztig variety for $z$ over the regular semisimple locus of $G$. We relate the resulting formula to unipotent Lusztig varieties, giving a new geometric model for unicellular LLT polynomials. We then consider Laurent polynomials $\alpha_{\psi, G}^z$ indexed by irreducible characters $\psi$, encoding how our formula decomposes into ungraded characters arising from the Springer theory of $G$. From evidence in low rank, we conjecture that if $\psi$ is inflated from type $A$ in a particular way, then the nonzero coefficients of $\alpha_{\psi, G}^z$ are positive and unimodal. This offers an answer to a 1993 question of Haiman about generalizing a conjecture he posed for symmetric groups. We also prove that the matrix formed by the $\alpha_{\psi, G}^z$ is partially triangular, and that their positivity and unimodality properties are stable under inclusions of Levi subgroups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a geometric model for unicellular LLT polynomials using closed Lusztig varieties and conjectures positive unimodal coefficients for certain α polynomials as a partial answer to Haiman's 1993 question, with clean proofs on triangularity and Levi stability but thin evidence for the main claim.

read the letter

The main point to know is that the paper constructs a geometric model for unicellular LLT polynomials by working with closed Lusztig varieties and their intersection cohomology, and it proposes a conjecture that the nonzero coefficients of the α_ψ,G^z polynomials are positive and unimodal when ψ is inflated from type A. This is presented as a response to Haiman's 1993 question on generalizing his conjecture from symmetric groups. The authors use results from Lusztig and Abreu-Nigro to find an explicit formula for the graded W-character over the regular semisimple locus. They then connect this to unipotent Lusztig varieties to obtain the model for the LLT polynomials. From there, they define the Laurent polynomials α that encode the decomposition into ungraded characters from Springer theory. They do a good job proving that the matrix assembled from these α polynomials is partially triangular. They also show that the positivity and unimodality properties remain stable when passing to Levi subgroups. These arguments seem to be based on straightforward computations within the given framework and hold up as stated. Where it is softer is in the conjecture. The support comes from low-rank evidence, and there is no general derivation or additional checks like error analysis. The stress test raises a fair point about whether the Lusztig-Abreu-Nigro theorems apply directly without extra hypotheses on the variety or stratification in this specific setting. If they do not, then the character formula and everything built on it would need adjustment. The paper treats the conjecture as such, but the evidential base is narrow. This paper is for people deep in geometric representation theory, particularly those interested in Springer representations, Lusztig varieties, and connections to algebraic combinatorics like LLT polynomials. A reader who knows the background on Haiman's conjecture and the relevant prior work would see the value in the new model and the stability results. It deserves a serious referee because it engages honestly with an old open question, supplies a fresh geometric perspective, and includes some clean partial theorems that can be checked. I recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper computes the graded W-character of the intersection cohomology of closed Lusztig varieties for a Weyl group element z over the regular semisimple locus of a reductive group G, invoking results of Lusztig and Abreu-Nigro. It relates this to unipotent Lusztig varieties to give a geometric model for unicellular LLT polynomials. It defines Laurent polynomials α_ψ,G^z that encode the decomposition into ungraded Springer characters, conjectures that these have positive unimodal coefficients when ψ is inflated from type A in a specified manner, and proves that the matrix of α_ψ,G^z is partially triangular with positivity/unimodality stable under Levi inclusions. This is positioned as a response to Haiman's 1993 question.

Significance. If the central conjecture holds and the cited theorems apply without additional hypotheses, the work supplies a new geometric model for unicellular LLT polynomials and a concrete generalization of Haiman's conjecture beyond symmetric groups, with the proved partial triangularity and Levi-stability providing immediate structural results in Springer theory. These elements would be of interest to researchers in geometric representation theory and combinatorial positivity.

major comments (2)

[Section deriving the graded W-character formula (around the invocation of Lusztig-Abreu-Nigro)] The decomposition into α_ψ,G^z and the subsequent conjecture rest on the graded W-character formula obtained by applying Lusztig and Abreu-Nigro results to intersection cohomology of closed Lusztig varieties over the regular semisimple locus. The manuscript should explicitly verify that all hypotheses of those theorems (e.g., smoothness or stratification conditions) hold in this setting; without this check the formula, the decomposition, and the low-rank evidence for positivity/unimodality are not guaranteed to be reliable.
[Section stating the conjecture and low-rank evidence] The conjecture that nonzero coefficients of α_ψ,G^z are positive and unimodal for type-A-inflated ψ is supported only by low-rank computations. Because this is the load-bearing claim answering Haiman's question, the paper should either supply a general argument or at minimum include a systematic error analysis or bound on the rank range examined.

minor comments (2)

[Definition of α_ψ,G^z] Clarify the precise definition of 'inflated from type A in a particular way' for ψ, including how the inflation interacts with the Weyl group action.
[Geometric model section] The relation to unicellular LLT polynomials via unipotent Lusztig varieties would benefit from an explicit comparison table or example in low rank.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestions. The comments have prompted us to strengthen the foundational justifications and the presentation of the computational evidence. We respond to each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Section deriving the graded W-character formula (around the invocation of Lusztig-Abreu-Nigro)] The decomposition into α_ψ,G^z and the subsequent conjecture rest on the graded W-character formula obtained by applying Lusztig and Abreu-Nigro results to intersection cohomology of closed Lusztig varieties over the regular semisimple locus. The manuscript should explicitly verify that all hypotheses of those theorems (e.g., smoothness or stratification conditions) hold in this setting; without this check the formula, the decomposition, and the low-rank evidence for positivity/unimodality are not guaranteed to be reliable.

Authors: We agree that an explicit verification of the hypotheses is required to ensure the applicability of the Lusztig and Abreu-Nigro theorems. In the revised manuscript we have added a new paragraph immediately after the statement of the graded W-character formula. This paragraph confirms that the closed Lusztig varieties over the regular semisimple locus are smooth (by the standard properties of Lusztig varieties recalled in Section 2) and that the G-orbit stratification satisfies the necessary conditions for the intersection-cohomology computations, citing the relevant results from Lusztig’s work on character sheaves. This verification guarantees that the formula, the decomposition into α_ψ,G^z, and the subsequent low-rank checks are rigorously justified in the present setting. revision: yes
Referee: [Section stating the conjecture and low-rank evidence] The conjecture that nonzero coefficients of α_ψ,G^z are positive and unimodal for type-A-inflated ψ is supported only by low-rank computations. Because this is the load-bearing claim answering Haiman's question, the paper should either supply a general argument or at minimum include a systematic error analysis or bound on the rank range examined.

Authors: The conjecture on positivity and unimodality is indeed central to our proposed answer to Haiman’s question, yet the manuscript’s primary results are the geometric model for unicellular LLT polynomials and the proved partial triangularity together with Levi stability. In the revision we have added a systematic error analysis: we derive an explicit upper bound on the possible degrees of the Laurent polynomials α_ψ,G^z from the dimension of the Lusztig varieties and the grading induced by the Springer action, and we extend the computational verification to all irreducible characters ψ inflated from type A in Weyl groups of rank at most 4. No counterexamples appear in this range. A general proof of the conjecture remains open and is stated as such in the paper. revision: partial

standing simulated objections not resolved

A general proof of the positivity and unimodality conjecture for the nonzero coefficients of α_ψ,G^z when ψ is inflated from type A.

Circularity Check

0 steps flagged

No significant circularity; external citations and low-rank evidence keep derivation self-contained

full rationale

The paper's central computation of the graded W-character of intersection cohomology for closed Lusztig varieties is obtained by direct invocation of prior theorems of Lusztig and Abreu-Nigro rather than derived internally from the paper's own equations. The polynomials α_ψ,G^z are then defined by decomposing this externally supplied formula into ungraded Springer characters. The main conjecture on positivity and unimodality for type-A-inflated ψ rests on separate low-rank computational checks, not on any algebraic identity forced by the paper's definitions. The proved statements (partial triangularity of the α-matrix and stability under Levi inclusions) are established by direct arguments within the manuscript. No self-definitional reductions, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the cited results are independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard facts from algebraic geometry and representation theory plus the specific theorems of Lusztig and Abreu-Nigro; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of intersection cohomology and graded characters for varieties over the regular semisimple locus hold as in the cited works.
Invoked to obtain the graded W-character formula.

pith-pipeline@v0.9.0 · 5741 in / 1438 out tokens · 43513 ms · 2026-05-20T03:14:12.638809+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use work of Lusztig and Abreu-Nigro to compute the graded W-character of the intersection cohomology of any closed Lusztig variety for z over the regular semisimple locus of G. ... τz,v = ∑ χ,ψ {χ,ψ} ψv(cz) χ ... αz_ψ,G(v) such that τz,v = ∑ ψ αz_ψ,G(v) sprψ,G
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture B: if ψ is inflated from a quotient φ:W→Ω with Ω a product of symmetric groups ... then nonzero coefficients of αz_ψ,G are positive and unimodal

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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[1] [1]

[AA08] P. N. Achar and A.-M. Aubert. Springer correspondences for dihedral groups.Transform. Groups, 13(1):1–24, 2008.doi:10.1007/s00031-008-9004-2. [Ach09] Pramod N. Achar. Springer theory for complex reflection groups. InRIMS Kôkyûroku

work page doi:10.1007/s00031-008-9004-2 2008

[2] [2]

kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1647.html

URL:https://www.kurims. kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1647.html. [Ach21] Pramod N. Achar.Perverse sheaves and applications to representation theory, volume 258 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, [2021]©2021.doi:10.1090/surv/258. 36 MINH-TÂM QUANG TRINH [AGV+73] M. Artin, A. Grothendieck, J. L. V...

work page doi:10.1090/surv/258 2021

[3] [3]

Notes Math.Springer, Cham, 1973.doi:10.1007/BFb0070714

Exposés IX à XIX, volume 305 ofLect. Notes Math.Springer, Cham, 1973.doi:10.1007/BFb0070714. [AN25] Alex Abreu and Antonio Nigro. A geometric approach to characters of Hecke algebras. J. Reine Angew. Math., 821:53–114, 2025.doi:10.1515/crelle-2024-0098. [AP18] Per Alexandersson and Greta Panova. LLT polynomials, chromatic quasisymmetric functions and grap...

work page doi:10.1007/bfb0070714 1973

[4] [4]

[BHL25] Patrick Brosnan, Jaehyun Hong, and Donggun Lee

doi:10.1016/j.aim.2018.02.020. [BHL25] Patrick Brosnan, Jaehyun Hong, and Donggun Lee. Geometry of regular semisimple Lusztig varieties,

work page doi:10.1016/j.aim.2018.02.020 2018

[5] [5]

[Car93] Roger W

v1.arXiv:2504.15868. [Car93] Roger W. Carter.Finite groups of Lie type. Wiley Classics Library. John Wiley & Sons, Ltd., Chichester,

work page arXiv

[6] [6]

[CHSS16] Samuel Clearman, Matthew Hyatt, Brittany Shelton, and Mark Skandera

Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication. [CHSS16] Samuel Clearman, Matthew Hyatt, Brittany Shelton, and Mark Skandera. Evalua- tions of Hecke algebra traces at Kazhdan-Lusztig basis elements.Electron. J. Comb., 23(2):research paper p2.7, 56,

work page 1985

[7] [7]

[CM18] Erik Carlsson and Anton Mellit

URL: www.combinatorics.org/ojs/index.php/ eljc/article/view/v23i2p7. [CM18] Erik Carlsson and Anton Mellit. A proof of the shuffle conjecture.J. Am. Math. Soc., 31(3):661–697, 2018.doi:10.1090/jams/893. [Del77] Pierre Deligne. Théoremes de finitude en cohomologieℓ-adique (avec un appendice par L. Illusie). Semin. Geom. algebr. Bois-Marie, SGA 4 1/2, Lect....

work page doi:10.1090/jams/893 2018

[8] [8]

On special pieces in the unipotent variety.Exp

[GM99] Meinolf Geck and Gunter Malle. On special pieces in the unipotent variety.Exp. Math., 8(3):281–290, 1999.doi:10.1080/10586458.1999.10504405. [GMR+25] SeanT.Griffin, AntonMellit, MarinoRomero, KevinWeigl, andJoshuaJeishingWen.On Macdonald expansions ofq-chromatic symmetric functions and the Stanley-Stembridge Conjecture,

work page doi:10.1080/10586458.1999.10504405 1999

[9] [9]

[Hai93] Mark Haiman

v1.arXiv:2504.06936. [Hai93] Mark Haiman. Hecke algebra characters and immanant conjectures.J. Am. Math. Soc., 6(3):569–595, 1993.doi:10.2307/2152777. [He25] Xuhua He. On affine Lusztig varieties.Ann. Sci. Éc. Norm. Supér. (4), 58(3):749–776, 2025.doi:10.24033/asens.2615. [Hik25] Tatsuyuki Hikita. A proof of the Stanley–Stembridge conjecture,

work page doi:10.2307/2152777 1993

[10] [10]

[Kat24] Syu Kato

v2.arXiv: 2410.12758. [Kat24] Syu Kato. A geometric realization of the chromatic symmetric function of a unit interval graph,

work page arXiv

[11] [11]

[Kim18] Dongkwan Kim

v2.arXiv:2410.12231. [Kim18] Dongkwan Kim. On total Springer representations for classical types.Sel. Math., New Ser., 24(5):4141–4196, 2018.doi:10.1007/s00029-018-0438-7. HAIMAN’S CONJECTURE AND SPRINGER’S REPRESENTATIONS 37 [KL79] David Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras.Invent. Math., 53:165–184, 1979.doi:1...

work page doi:10.1007/s00029-018-0438-7 2018

[12] [12]

Princeton University Press, Princeton, NJ, 1984.doi:10.1515/ 9781400881772

[Lus84] George Lusztig.Characters of reductive groups over a finite field, volume 107 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1984.doi:10.1515/ 9781400881772. [Lus85a] George Lusztig. Character sheaves. II.Adv. Math., 57:226–265, 1985.doi:10.1016/ 0001-8708(85)90064-7. [Lus85b] George Lusztig. Character sheaves. III.Adv...

work page doi:10.1007/bf02699129 1984

[13] [13]

[Lus97] G

With an appendix by Gunter Malle.doi:10.1215/S0012-7094-94-07309-2. [Lus97] G. Lusztig. Notes on unipotent classes.Asian J. Math., 1(1):194–207, 1997.doi:10. 4310/AJM.1997.v1.n1.a7. [Lus11] George Lusztig. On some partitions of a flag manifold.Asian J. Math., 15(1):1–8,

work page doi:10.1215/s0012-7094-94-07309-2 1997

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